Advertisement

Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 103–132 | Cite as

Riemannian metrics on differentiable stacks

  • Matias del HoyoEmail author
  • Rui Loja Fernandes
Article

Abstract

We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.

Notes

Acknowledgements

We are grateful to IMPA, UU and UIUC for hosting us at several stages of this project. We thank H. Bursztyn, E. Lerman, I. Marcut and I. Moerdijk for fruitful discussions, and to M. Crainic, J.N. Mestre and I. Struchiner for sharing with us a preliminary version of their preprint [6]. We also thank the referee for his comments and suggestions, that helped improve this manuscript.

References

  1. 1.
    Arias Abad, C., Crainic, M.: Representations up to homotopy and Bott’s spectral sequence for Lie groupoids. Adv. Math. 248, 416–452 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Artin, M., Grothendieck, A., Verdier, J.L.: Thorie des topos et cohomologie tale des schmas (SGA 4-2). Lecture Notes in Mathematics 270. Springer, New York (1972)Google Scholar
  3. 3.
    Behrend, K., Xu, P.: Differentiable stacks and gerbes. J Symplectic Geom 9, 285–341 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bursztyn, H., Cabrera, A., del Hoyo, M.: Vector bundles over Lie groupoids and algebroids. Adv. Math. 290, 163–207 (2016). arXiv:1410.5135 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bursztyn, H., Noseda, F., Zhu, C.: Principal actions of stacky Lie groupoids. Preprint arXiv:1510.09208
  6. 6.
    Crainic, M., Mestre, J.N., Struchiner, I.: Deformations of Lie Groupoids. Preprint arXiv:1510.02530
  7. 7.
    Crainic, M., Struchiner, I.: On the Linearization Theorem for proper Lie groupoids. Ann. Scient. Éc. Norm. Sup. \(4^e\) série 46, 723–746 (2013)Google Scholar
  8. 8.
    del Hoyo, M.: On the homotopy type of a cofibred category. Cahiers de Topologie et Geometrie Differentielle Categoriques 53, 82–114 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    del Hoyo, M.: Lie groupoids and their underlying orbispaces. Portugaliae Mathematica 70, 161–209 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    del Hoyo, M., de Melo, M.: Geodesics on Differentiable Stacks. Work in progressGoogle Scholar
  11. 11.
    del Hoyo, M., Fernandes, R.L.: Riemannian Metrics on Lie Groupoids. J. für die reine und angewandte Mathematik (Crelle), (2015). Preprint arXiv:1404.5989
  12. 12.
    del Hoyo, M., Fernandes, R.L.: On deformations of compact foliations. Preprint arXiv:1807.10748
  13. 13.
    Epstein, D., Rosenberg, H.: Stability of compact foliations. In: Geometry and Topology. Lecture Notes in Mathematics, vol. 597, pp. 151–160. Springer (1977)Google Scholar
  14. 14.
    Giraud, J.: Cohomologie non-abélienne; Die Grundlehren der mathematischen Wissenschaften, vol. 179. Springer, Berlin (1971)Google Scholar
  15. 15.
    Hamilton, R.: Deformation theory of foliations. Available from Cornell University in mimeographed formGoogle Scholar
  16. 16.
    Lerman, E.: Orbifolds as stacks? L’Enseign. Math. (2) 56(3–4), 315–363 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Mackenzie, K.: General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series, vol. 213. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  18. 18.
    Metzler, D.: Topological and Smooth Stacks. Preprint arXiv:math/0306176
  19. 19.
    Moerdijk, I., Mrcun, J.: Introduction to Foliations and Lie Groupoids. Cambridge Studies in Advanced Mathematics, vol. 91. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  20. 20.
    Moerdijk, I., Mrcun, J.: Lie groupoids, sheaves and cohomology. Lond. Math. Soc. Lect. Note Ser. 323, 145–272 (2005)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Nijenhuis, A., Richardson, R.: Deformations of homomorphisms of Lie groups and Lie algebras. Bull. Am. Math. Soc. 73(1), 175–179 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Palais, R., Richardson, R.: Uncountably many inequivalent analytic actions of a compact group on \({\mathbb{R}}^n\). Proc. Am. Math. Soc. 14(3), 374–377 (1963)zbMATHGoogle Scholar
  23. 23.
    Palais, R., Stewart, T.: Deformations of compact differentiable transformation groups. Am. J. Math. 82(4), 935–937 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Vistoli, A.: Grothendieck topologies, fibered categories and descent theory; Fundamental algebraic geometry, 1104, Math. Surveys Monogr., 123, AMS Providence, RI (2005)Google Scholar
  25. 25.
    Weinstein, A.: Linearization of regular proper groupoids. J. Inst. Math. Jussieu 1(3), 493–511 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zung, N.T.: Proper groupoids and momentum maps: linearization, affinity, and convexity. Ann. Sci. École Norm. Sup. (4) 39(5), 841–869 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Geometria - IMEUniversidade Federal FluminenseNiteróiBrazil
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations